Theorem amosym1 32425

Description: A symmetry with ∃*.

See negsym1 32416 for more information. (Contributed by Anthony Hart, 13-Sep-2011.)

Assertion

Ref	Expression
amosym1	⊢ (∃*𝑥∃*𝑥⊥ → ∃*𝑥𝜑)

Proof of Theorem amosym1

Step	Hyp	Ref	Expression
1		df-mo 2475	. 2 ⊢ (∃*𝑥∃*𝑥⊥ ↔ (∃𝑥∃*𝑥⊥ → ∃!𝑥∃*𝑥⊥))
2		mof 32409	. . . . 5 ⊢ ∃*𝑥⊥
3		19.8a 2052	. . . . . 6 ⊢ (∃*𝑥⊥ → ∃𝑥∃*𝑥⊥)
4	3	notnotd 138	. . . . 5 ⊢ (∃*𝑥⊥ → ¬ ¬ ∃𝑥∃*𝑥⊥)
5	2, 4	ax-mp 5	. . . 4 ⊢ ¬ ¬ ∃𝑥∃*𝑥⊥
6	5	pm2.21i 116	. . 3 ⊢ (¬ ∃𝑥∃*𝑥⊥ → ∃*𝑥𝜑)
7	2	notnoti 137	. . . . . 6 ⊢ ¬ ¬ ∃*𝑥⊥
8	7	nex 1731	. . . . 5 ⊢ ¬ ∃𝑥 ¬ ∃*𝑥⊥
9		eunex 4859	. . . . 5 ⊢ (∃!𝑥∃*𝑥⊥ → ∃𝑥 ¬ ∃*𝑥⊥)
10	8, 9	mto 188	. . . 4 ⊢ ¬ ∃!𝑥∃*𝑥⊥
11	10	pm2.21i 116	. . 3 ⊢ (∃!𝑥∃*𝑥⊥ → ∃*𝑥𝜑)
12	6, 11	ja 173	. 2 ⊢ ((∃𝑥∃*𝑥⊥ → ∃!𝑥∃*𝑥⊥) → ∃*𝑥𝜑)
13	1, 12	sylbi 207	1 ⊢ (∃*𝑥∃*𝑥⊥ → ∃*𝑥𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ⊥wfal 1488  ∃wex 1704  ∃!weu 2470  ∃*wmo 2471

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-nul 4789  ax-pow 4843

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-eu 2474  df-mo 2475

This theorem is referenced by: (None)